In the product $\vec{F} = q(\vec{v} imes \vec{B}) = q\vec{v} imes (B\hat{i} + B\hat{j} + B_0\hat{k})$. For

Question

In the product $\vec{F} = q(\vec{v} 	imes \vec{B}) = q\vec{v} 	imes (B\hat{i} + B\hat{j} + B_0\hat{k})$. For $q=1$ and $\vec{v} = 2\hat{i} + 4\hat{j} + 6\hat{k}$ and $\vec{F} = 4\hat{i} - 20\hat{j} + 12\hat{k}$, what will be the complete expression for $\vec{B}$?

Accepted Answer

1.  **Identify Formula:** The force on a moving charge is given by the Lorentz force equation $\vec{F} = q(\vec{v} 	imes \vec{B})$.
2.  **Substitute Knowns:**
    *   $q = 1$
    *   $\vec{v} = 2\hat{i} + 4\hat{j} + 6\hat{k}$
    *   $\vec{F} = 4\hat{i} - 20\hat{j} + 12\hat{k}$
    *   $\vec{B}$ is assumed to be of the form $B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$. The problem states $\vec{B} = B\hat{i} + B\hat{j} + B_0\hat{k}$, implying $B_x = B_y$.
3.  **Evaluate Options:** We can test the options by calculating the cross product $\vec{v} 	imes \vec{B}$ for each.
    *   **Test Option D:** $\vec{B} = -6\hat{i} - 6\hat{j} - 8\hat{k}$
    $$\vec{v} 	imes \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 4 & 6 \ -6 & -6 & -8 \end{vmatrix}$$
    *   $\hat{i}$-component: $(4)(-8) - (6)(-6) = -32 + 36 = 4$
    *   $\hat{j}$-component: $-[(2)(-8) - (6)(-6)] = -[-16 + 36] = -20$
    *   $\hat{k}$-component: $(2)(-6) - (4)(-6) = -12 + 24 = 12$
    *   Result: $4\hat{i} - 20\hat{j} + 12\hat{k}$.
4.  **Conclusion:** The calculated force matches the given force $\vec{F}$. Therefore, the magnetic field vector is $\vec{B} = -6\hat{i} - 6\hat{j} - 8\hat{k}$.

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